3-Forms, Axions and D-Brane Instantons

3-Forms, Axions and D-brane Instantons

Eduardo Garc´ıa-Valdecasas Tenreiro

Instituto de F´ısica Teorica´ UAM/CSIC, Madrid

Based on 1605.08092 by E.G. & A. Uranga

V Postgradute Meeting on Theoretical Physics, Oviedo, 18th November 2016 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Motivation & Outline

Motivation Axions have very ﬂat potentials ⇒ Good for Inﬂation. Axions with non-perturbative (instantons) potential can always be described as a 3-Form eating up the 2-Form dual to the axion. There is no candidate 3-Form for stringy instantons. We will look for it in the geometry deformed by the instanton. Outline

1 Axions, Monodromy and 3-Forms. 2 String Theory, D-Branes and Instantons. 3 Backreacting Instantons.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 2 / 20 2 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Axions and Monodromy.

Axions are periodic scalar ﬁelds.

This shift symmetry can be broken to a discrete symmetry by non-perturbative effects → Very ﬂat potential, good for inﬂation.

The discrete shift symmetry gives periodic potentials, but non-periodic ones can be used by endowing them with a monodromy structure. For instance: |φ|2 + µ2φ2 (1)

Superplanckian ﬁeld excursions in Quantum Gravity are under pressure due to the Weak Gravity Conjecture (Arkani-Hamed et al., 2007). Monodromy may provide a workaround (Silverstein & Westphal, 2008; Marchesano et al., 2014).

This mechanism can be easily realized in string theory.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 3 / 20 3 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Kaloper-Sorbo Monodromy

A quadratic potential with monodromy can be described by a Kaloper-Sorbo lagrangian (Kaloper & Sorbo, 2009),

2 2 | dφ| + nφF4 + |F4| (2)

Where the monodromy is given by the vev of F4, different ﬂuxes correspond to different branches of the potential.

(Ibanez et al., 2016) Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 4 / 20 4 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Kaloper-Sorbo Monodromy

Monodromy structure visible in potential solving E.O.M,

2 V0 ∼ (nφ − q) , q ∈ Z (3)

So there is a discrete shift symmetry,

φ → φ + φ0, q → q + nφ0 (4)

A dual description can be found using the hodge dual of φ, b2,

2 2 | db2 + nc3| + |F4| , F4 = dc3 (5)

The ﬂatness of the axion potential is protected by gauge invariance,

c3 → c3 + dΛ2 , b2 → b2 − nΛ2 (6)

We will look for a three form c3 coupling to the axion as ∼ φF4 This description has the periodicity built in. Whenever the potential breaks it, there will be monodromy in the UV.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 5 / 20 5 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Example: Peccei-Quinn mechanism in 3-Form language.

Strong CP problem: QCD has an anomalous U(1)A symmetry producing a physical θ-term.

g2θ L ∼ F a F˜ aµν (7) 32π2 µν

Where θ classiﬁes topologically inequivalent vacua and is typically ∼ 1. This term breaks CP and experimental data −10 constrain θ . 10 → Fine Tuning. Solution: PQ Mechanism. New anomalous spontaneously

broken U(1)PQ. So θ is promoted to pseudo-Goldstone boson, the axion. Non-perturbative effects give it a potential and ﬁx θ = 0.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 6 / 20 6 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Example: Peccei-Quinn mechanism in 3-Form language.

The term can be rewritten as (Dvali, 2005),

g2 3 θFF˜ ∼ θF = θ dC , C = Tr A A A − A ∂ A 4 3 αβγ 8π2 [α β γ] 2 [α β γ] (8)

So, making θ small = How can I make the F4 electric ﬁeld small?

Solution: Screen a ﬁeld ⇒ Higgs mechanism. Let C3 eat a

2-Form b2! ⇒ b2 is Hodge dual of axion! Potential for the axion ⇐⇒ 3-Form eating up a 2-form.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 7 / 20 7 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons String Theory

String theory unifies gravity and quantum mechanics. String theory aims to describe everything as vibrations of tiny strings (see IFT youtube). → Each particle is a different vibration state of the string ⇒ So string theory is awesome! Caveats: 10 dimensions, infinite (or very big) landscape of vacua, only pertubatively defined... But who cares?

→ Gravity is described by closed strings and gauge interactions by open strings.

→ 6 dimensions are compact: M4 × X 6

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 8 / 20 8 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons D-Branes

Open Strings end on non-perturbative, dynamical objects called D-Branes,

→ Rich physcis inside the Brane.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 9 / 20 9 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons D-Brane Instantons

Dp-Branes completely wrapped around euclidean p cycles are D-brane instantons. There are two kinds: Dp-Brane instanton inside D(p + 4)-Brane → Gauge Instanton. Dp-Brane alone → Stringy Instanton. ⇒ We study non-perturbative potentials for axions coupling to stringy instantons. There are 5 string theories, all related through dualities. We will focus on type IIB. It has odd Dp-branes and odd RR Forms:

F1, F3, F5...

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 10 / 20 10 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Axions in String Theory.

Axions are ubiquitous in string theory. For instance upon compactiﬁcation, Z a(x) = Cp (9) Σp Where the shift symmetry arises from the gauge invariance of the RR form. Monodromy can be realised in many ways in string theory. For instance, as unwinding of a brane (Silverstein & Westphal, 2008),

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 11 / 20 11 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons The puzzle

Axion potentials come from non-perturbative effects and can be described by a 3-form eating up a 2-form. For instance, for gauge D-brane instantons it is the CS 3-Form (Dvali, 2005).

For stringy instantons → No candidate 3-Form! ⇒ Solution: look for the 3-Form in the backreacted geometry (Koerber & Martucci, 2007). In the backreacted geometry the instanton disappears and its open-string degrees of freedom are encoded into the geometric (closed strings) degrees of freedom. Both descriptions are related in an ”holographic” way.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 12 / 20 12 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons SU(3) × SU(3) structure manifolds.

The backreacted geometry will, generically, be a non CY, SU(3) × SU(3) (1) (2) structure manifold. This structure has two globally well deﬁned spinors η+ , η+ , (1) (2) with c.c. η− , η− . Deﬁne two polyforms (assume type IIB):

i X 1 (2)† (1) m m Ψ± = − η γm ...m η dy l ∧ ... ∧ dy 1 (10) ||η(1)||2 l! ± 1 l + l

Organize 10d ﬁelds in holomorphic polyforms,

3A−Φ −Φ Z ≡ e Ψ2, T ≡ e Re Ψ1 + i∆C (11)

For an SU(3) structure manifold, η(1) ∼ η(2) and one recovers Z ∼ Ω, T ∼ eiJ

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 13 / 20 13 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Backreacting a D3-instanton

Consider type IIB String Theory with a D3-instanton wrapping a

4-cycle Σ4 in a CY3. One can show from the SUSY conditions for the D3-instanton that the backreaction is encoded in a contribution to Z (Koerber & Martucci, 2007):

d(δZ) ∼ Wnpδ2(Σ4) (12)

⇒ So, the backreaction produces a 1-form Z1 that didn’t exist in the original geometry.

Note that Z1 is not closed and thus not harmonic.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 14 / 20 14 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons The 3-form and the KS coupling

Let us deﬁne α1 ≡ Z1 and β2 ≡ dα1.

In type IIB string theory there is a RR 4-form C4. We may expand it as,

C4 = α1(y) ∧ c3(x) + β2(y) ∧ b2(x) + ... (13)

⇒ So we have a 3-form and a 2-form dual to an scalar.

We see that,

F5 = dC4 = β2 ∧ (c3 + db2) − α1 ∧ F4 (14) Which describes a 3-Form eating up a 2-Form, as we wanted! Furthermore, Z Z Z Z F5 ∧ ∗F5 = − C4 ∧ dF5 → C4 ∧ β2 ∧ F4 = φF4 (15) 10d 10d 10d 4d Which is the KS coupling we were looking for.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 15 / 20 15 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Toroidal examples.

˜ For a 4-cycle deﬁned by the equation f = 0 the 1-form is Z(1) ∼ df Wnp. Example 1: Factorisable 6-torus, M4 × T2 × T2 × T2 with local

coordinates z1, z2, z3 and take the cycle f = z3 = 0,

−T Z(1) ∼ e dz3 (16)

⇒ already in the original geometry, because T6 is non-CY.

6 Example 2: Orbifold T /(Z2 × Z2) with action,

θ : (z1, z2, z3) → (−z1, z2, −z3), ω : (z1, z2, z3) → (z1, −z2, −z3) (17)

2 Local coordinates ui = zi and cycle f ∼ u3 = 0 give backreaction,

Z(1) ∼ du3 ∼ z3 dz3 (18)

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 16 / 20 16 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons D-brane gauge Instanton backreaction

D3-brane instantons wrapping the same 4-cycle as N spacetime-ﬁlling D7-branes are gauge instantons from the SU(N) point of view.

In the gauge (open string) description a CS 3-Form is available to couple to the axion.

We can backreact the D7’s together with their non-perturbative effects to ﬁnd a dual description in terms of closed string degrees of freedom. Again a 1-Form arises when taking the non-perturbative effects into account,

dZ = ils hSi δ2(Σ) (19)

Where S is the gaugino condensate describing the non-perturbative effects.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 17 / 20 17 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Generalization

Supersymmetry equations in type IIA are trickier, but we can use mirror symmetry. Mirror symmetry is a symmetry between IIA and IIB living in

mirror CY3’s. In the large complex structure limit it equal to 3 T-dualities. The mirror dual of our setup consists on a D2-brane wrapping a 3-cycle. The backreaction gives rise to,

δT = T(2) + T(4) (20)

⇒ so we can obtain the 3-Form in two ways,

C7 = T(4) ∧ c3, C5 = T(2) ∧ c3 (21)

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 18 / 20 18 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons Conclusions & Outlook

Axions are a useful tool for inﬂation and are easily realised in String Theory.

Axion Monodromy may help avoid the Weak Gravity Conjecture.

Axion physics can be described in a dual language where a 3-form eats up a 2-form.

For stringy instantons no known 3-form to couple to the 2-form was known. We have showed that it only appears when the ”backreaction” is taken into account.

The geometry ceases to be CY and new forms (and cycles) that were not there arise.

Further work:

Use this mechanism as a self consistency test for the Weak Gravity Conjecture for axions and 3-Forms. Study the backreaction of particular instantons in speciﬁc setups ⇒ ⇒ Geometric transitions.

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 19 / 20 19 / 20 Axions, Monodromy and 3-Forms String Theory, D-Branes and Instantons Backreacting Instantons

Thank You!

Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 20 / 20 20 / 20 References

Arkani-Hamed, Nima, Motl, Lubos, Nicolis, Alberto, & Vafa, Cumrun. 2007. The String landscape, black holes and gravity as the weakest force. JHEP, 06, 060. Dvali, Gia. 2005. Three-form gauging of axion symmetries and gravity. Ibanez, Luis E., Montero, Miguel, Uranga, Angel, & Valenzuela, Irene. 2016. Relaxion Monodromy and the Weak Gravity Conjecture. JHEP, 04, 020. Kaloper, Nemanja, & Sorbo, Lorenzo. 2009. A Natural Framework for Chaotic Inﬂation. Phys. Rev. Lett., 102, 121301. Koerber, Paul, & Martucci, Luca. 2007. From ten to four and back again: How to generalize the geometry. Eduardo Garc´ıa-Valdecasas Tenreiro 3-Forms, Axions and D-brane Instantons 20 / 20 20 / 20 References

JHEP, 08, 059.

Marchesano, Fernando, Shiu, Gary, & Uranga, Angel M. 2014. F-term Axion Monodromy Inﬂation. JHEP, 09, 184.

Silverstein, Eva, & Westphal, Alexander. 2008. Monodromy in the CMB: Gravity Waves and String Inﬂation. Phys. Rev., D78, 106003.

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